Exercises: Distances in the plane

Calculate the exact distance between the points $A(3, -2)$ and $B(-1, 1)$.

Find the distance from the origin to the line with equation $3x - 4y + 10 = 0$.

Determine the distance from the point $P(2, 5)$ to the line $r: 5x + 12y - 4 = 0$.

Calculate the distance from the point $Q(-1, 3)$ to the line given in explicit form $y = 2x - 1$.

Find the distance separating the parallel lines $r: 3x + 4y - 5 = 0$ and $s: 3x + 4y + 15 = 0$.

Check whether the triangle with vertices $A(0, 0)$, $B(4, 0)$ and $C(2, 3)$ is isosceles by calculating the exact length of its three sides.

Calculate the distance from the point $P(-4, 7)$ to the y-axis and to the x-axis.

Given the segment with endpoints $A(2, 4)$ and $B(6, -2)$, first calculate the coordinates of its midpoint $M$, and then find the distance from $M$ to the endpoint $A$.

A surveying team has marked out a triangular plot of land whose base stations are located at coordinates $A(1, 2)$, $B(5, -1)$ and $C(4, 6)$, expressed in kilometers. Calculate the exact area of the land, first evaluating the length of the base $AB$ and then the corresponding height from the vertex $C$.

A telecommunications antenna is to be installed on a straight road that coincides with the x-axis. This antenna must be equidistant from village $P(-2, 3)$ and village $Q(4, 1)$. What exact coordinates should the antenna be placed at?

Two ships follow straight, parallel routes. The first ship sails along the line $r: x - 2y + 4 = 0$. The second ship passes through the point $A(3, 1)$ maintaining the same heading. Calculate the minimum distance separating the two paths.

In a park, the center of a square is located at the point $P(0, 5)$. A straight irrigation pipe follows the path $y = x + 1$. A sprinkler is to be installed on the pipe at a distance of exactly $5$ units from the center of the square. Find the coordinates of the two possible locations of the sprinkler.

An industrial laser needs to make a main cut in a metal sheet along the line $r: 4x - 3y + 2 = 0$. In addition, the system must be programmed to make two parallel safety cuts at an exact distance of $10$ mm on each side of the main cut. Find the general equations of the lines for the safety cuts.

A space exploration vehicle moves in a straight line following the equation $y = 2x$. An emergency wireless charging base is located at the point $C(5, 0)$. Determine at which exact point on its trajectory the vehicle will be at the minimum distance from the base, and calculate that distance.

A solar panel is in the shape of a quadrilateral with vertices at the coordinates $A(-1, -1)$, $B(3, 0)$, $C(4, 4)$ and $D(0, 3)$, measured in meters. Calculate the exact perimeter of the panel.

A comet passes through a star system following a straight line trajectory with equation $3x + y - 6 = 0$. An observation satellite is stationary at the coordinates $S(2, 4)$. Calculate the minimum distance at which the comet will pass the satellite.

Find the possible values of the parameter $k$ such that the distance from the line $r: kx + 3y - 5 = 0$ to the point $P(1, -2)$ is exactly $2$ units.

The lines $r: x - 2y + 1 = 0$ and $s: 2x + y - 3 = 0$ represent the edges of two intersecting avenues. A cycle lane is to be built that is always equidistant from both avenues. Find the equations of the two possible paths for the cycle lane (the angle bisectors) and show analytically that they are perpendicular.

An isosceles triangle has its unequal side (the base) on the line $r: x + y - 2 = 0$. The vertex opposite the base is $A(4, 4)$ and it is known that the area of the triangle is $12$ u$^2$. Calculate the exact length of the base and of the two equal sides.

Find the coordinates of the points in the plane that simultaneously satisfy two conditions: they are equidistant from the intersecting lines $r: 3x - 4y + 2 = 0$ and $s: 4x + 3y - 1 = 0$, and their distance from the origin is exactly $5$ units.

Determine the coordinates of the vertices of a square knowing that one of its vertices is $A(1, 2)$ and that one of its diagonals lies on the line $d: x - y - 1 = 0$.

Find the equation of the locus of points in the plane $P(x, y)$ whose distance to the line $r: 3x - 4y = 0$ is exactly twice their distance to the fixed point $Q(0, 2)$. Simplify the resulting equation.

Calculate the possible values of the parameter $c$, knowing that the distance between the parallel lines $r: 2x - y + c = 0$ and $s: 4x - 2y + 5 = 0$ is $3\sqrt{5}$ units.

A rhombus has one of its vertices at point $A(2, 1)$, and the diagonal that does not pass through $A$ lies on the line $r: x + 2y - 9 = 0$. If the area of the rhombus is $20$ u$^2$, find the coordinates of the remaining three vertices.